What is the criterion for a change to be adiabatic? I'm trying to understand whether the change of a parameter $\lambda$ of a Hamiltonian $H$ is adiabatic. Reading Landau and Lifshitz "Mechanics", I see

... let us suppose that $\lambda$ varies slowly (adiabatically) with time as the result of some external action; by a "slow" variation we mean one in which $\lambda$ varies only slightly during the period $T$ of the motion:
  $$T\frac{d\lambda}{dt}\ll\lambda.$$

But we can choose $\lambda$ in such a way that it has arbitrarily large value by taking Hamiltonian as $H'(\lambda)=H(\lambda-C)$, where $C=\mathrm{const}$ is a large compensating constant. Thus the condition of adiabaticity would be automatically fulfilled for arbitrarily fast change.
Reading Wikipedia, I see

In mechanics, an adiabatic change is a slow deformation of the Hamiltonian, where the fractional rate of change of the energy is much slower than the orbital frequency.

But again, we can shift energy by arbitrarily large constant without affecting the equations of motion, and then any change of energy will have very small logarithmic derivative.
Thus the criteria given above are too ambiguous to be usable.
So, what is the true unambiguous criterion for the change to be adiabatic? Or, if the criteria cited above are unambiguous, then what is my mistake?
 A: This is a good question and the answer is... we don't really know.
The adiabatic theorem can be stated in various forms. A rigorous one is the following. Assume the Hamiltonian changes in time only as a function of $t/T$, where $T$ is a timescale (total evolution time). Under a gap and other regularity conditions, if you start in an eigenstate of the Hamiltonian at time zero and evolve up to a time $t<T$ the adiabatic error is
$$
\frac{C}{T}.
$$
Where $C$ is a constant independent of $T$. Your question asks for a precise estimate of the constant $C$. This can be obtained in various ways, a rigorous estimate is provided in [this paper]( 
https://arxiv.org/abs/quant-ph/0603175)
by Jansen and Ruskai and Sailer. But this expression is not optimal and an optimal expression is not known. There is an "estimate" that is referred to in standard undergrad textbook which is usually called naiive condition. It is known not to be necessary actually but it provides a good order of magnitude. 
A: Look to the more fundamental, classical definition. Adiabatic just means 'without heat transfer'. But more specifically it requires defining a system or control volume where the heat is not transferred.
Your model or equations of motion must exist over some defined space and time interval. If heat is not transferred in/out of the system, the process is adiabatic.
